Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
| direct product | C2×#2015 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1083 conjugacy classes of subgroups; only the 231 normal subgroups are shown.
This group has 1083 conjugacy classes of subgroups — too many to draw. Only its 231 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(3,4)(5,6)(11,12,16,20)(13,21,24,25)(14,19,15,18)(17,23,26,22);(4,6)(7,8)(9,10)(11,13,16,24)(12,17,20,26)(14,23,15,22)(18,25,19,21);(8,10)(12,18)(14,15)(17,25)(19,20)(21,26)(22,23)64 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#2015, #359045, #359053, … |
| maximal normal subgroup of | C4×#2015, #359045, #359053, … |
| maximal quotient of | C2×#9965, C2×#9969, #232212, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16406);